The Witt group in the title of this paper is generated by equivalence classes of nondegenerate braided fusion categories, with a relation \([\mathcal{C}]=[\mathcal{D}]\) if \(\mathcal{C}\) and \(\mathcal{D}\) become equivalent after taking Deligne products with Drinfeld centers, that is, \(\mathcal{C}\boxtimes Z(\mathcal{A}_1)\cong \mathcal{D}\boxtimes Z(\mathcal{A}_2)\), where \(Z(\mathcal{A}_i)\) are Drinfeld centers of fusion categories \(\mathcal{A}_i\). The product in the Witt group is given by the Deligne product of fusion categories, and inverses are obtained by reversing the braiding of a braided fusion category.
Important examples \(\mathcal{C}(\mathfrak{g}, k)\) of (modular) braided fusion categories arise from representations of affine Lie algebras \(\widehat{\mathfrak{g}}\) at non-negative integer levels \(k\) ([M. Finkelberg, Geom. Funct. Anal. 6, No. 2, 249–267 (1996; Zbl 0860.17040); erratum ibid. 23, No. 2, 810–811 (2013)], [Y.-Z. Huang and J. Lepowsky, Duke Math. J. 99, No. 1, 113–134 (1999; Zbl 0953.17016)]), or equivalently, from representations of quantum groups \(U_q(\mathfrak{g})\) at suitable roots of unity \(q\) ([H. H. Andersen and J. Paradowski, Commun. Math. Phys. 169, No. 3, 563–588 (1995; Zbl 0827.17010)]).
In the paper under review, the author finds all Witt group relations between the equivalence classes \([\mathcal{C}(\mathfrak{sl}_3, k)]\), as well as all relations involving only the classes \([\mathcal{C}(\mathfrak{sl}_3, k)]\) and \([\mathcal{C}(\mathfrak{sl}_2, k)]\) (relations involving only the classes \([\mathcal{C}(\mathfrak{sl}_2, k)]\) were classified in [A. Davydov et al., Sel. Math., New Ser. 19, No. 1, 237–269 (2013; Zbl 1345.18005)]). The classification of Witt group relations is based on finding a category in each equivalence class \([\mathcal{C}(\mathfrak{sl}_3, k)]\) that has no non-trivial fusion subcategories or non-trivial connected étale algebras. For most cases where \(3\,|\,k\), the author shows that the braided tensor category of dyslectic modules for a certain maximal connected étale algebra in \(\mathcal{C}(\mathfrak{sl}_3, k)\) can be used; for the remaining cases, he uses a result of T. Gannon [Commun. Math. Phys. 161, No. 2, 233–263 (1994; Zbl 0806.17031)] that additional non-trivial connected étale algebras occur in the categories \(\mathcal{C}(\mathfrak{sl}_3,k)\) only for the levels \(k=5, 9, 21\).
The paper concludes with an overview of the obstructions to completing a similar classification of Witt group relations between classes \([\mathcal{C}(\mathfrak{g},k)]\) for higher-rank \(\mathfrak{g}\).